7.25.11 problem 11

Internal problem ID [631]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.4 (The eigenvalue method for homogeneous systems). Problems at page 378
Problem number : 11
Date solved : Wednesday, February 05, 2025 at 03:50:57 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 4 \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 23

dsolve([diff(x__1(t),t) = x__1(t)-2*x__2(t), diff(x__2(t),t) = 2*x__1(t)+x__2(t), x__1(0) = 0, x__2(0) = 4], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= -4 \,{\mathrm e}^{t} \sin \left (2 t \right ) \\ x_{2} \left (t \right ) &= 4 \,{\mathrm e}^{t} \cos \left (2 t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 26

DSolve[{D[x1[t],t]==x1[t]-2*x2[t],D[x2[t],t]==2*x1[t]+x2[t]},{x1[0]==0,x2[0]==4},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to -4 e^t \sin (2 t) \\ \text {x2}(t)\to 4 e^t \cos (2 t) \\ \end{align*}